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Channel Fading Models for 5G and 6G

Written By

Maryam Olyaee, Hadi Hashemi and Juan Manuel Romero Jerez

Reviewed: 07 March 2024 Published: 29 May 2024

DOI: 10.5772/intechopen.1005006

5G and 6G Enhanced Broadband Communications IntechOpen
5G and 6G Enhanced Broadband Communications Edited by Isiaka Ajewale Alimi

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5G and 6G Enhanced Broadband Communications

Edited by Isiaka Ajewale Alimi and Jide Julius Popoola

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Abstract

As communication technologies evolve, existing channel models need to be improved as new spectral bands, environments, and the use cases present propagation characteristics that are not properly captured by the well-known classical models. Small-scale signal fading, due to multipath propagation and the relative movement between transmitter and receiver, is one of the most important propagation effects in a wide range of wireless scenarios. Accurate modeling of this phenomenon in the millimeter-wave band has become particularly important due to its foreseen use in the 5G standard of cellular networks and beyond. In this chapter, we review several current channel models for new generations of wireless communication systems. These channel models consider different propagation mechanisms, such as specular (dominant) waves, multicluster reception, and signal fluctuations, and are studied based on their applications in both line-of-sight (LOS) and non-line-of-sight (NLOS) links.

Keywords

  • multi-clusters
  • small-scale fading
  • shadowing
  • line-of-sight (LOS)
  • non-line-of-sight (NLOS)
  • propagation
  • channel measurement
  • fluctuation
  • channel probability distribution

1. Introduction

Triggered by the emergence of new wireless communication scenarios, such as vehicular-to-vehicle, wireless sensor networks (WSNs), internet of things (IoT), and mmWave, the field of wireless channel modeling has received a new impetus. In this context, several new wireless channel models have emerged in the last few years. Moreover, there are new relevant results related to classical channel models proposed in the 1950s or 1960s. Although the channel propagation conditions for 5G millimeter-wave wireless communications are different from previous generations, some of the theoretical works on these networks still use the classical channel models, which may not lead to an accurate analysis. The heterogeneous distribution of scatterers, channel nonlinearities, independent reception of clusters of waves, and fluctuation of the dominant wave amplitudes, among other factors, lead to received power variations of the radio signal, which are not considered in earlier models.

Since the beginning of the twenty-first century, many new small-scale (multipath) fading models have been reported in the literature. A general model for the fading channel based on the reception of an arbitrary number of waves was introduced in 2002 by Rappaport et al. [1]. Due to the complexity of this model in its more general form, some specific cases have been studied.

In order to facilitate the usefulness of analytical stochastic wireless fading models, the following properties are desirable: (i) good fit to the propagation conditions: thus, advanced models usually have several parameters that can take different values depending on the propagation environment, allowing the model to adapt to different scenarios; (ii) physical interpretation: any random variable that fits well in a given propagation environment can ultimately be used to model the radio channel. However, this strategy does not provide information about how the wireless system under study might behave if the conditions change. Therefore, the parameters that characterize the model should have a clear physical interpretation that facilitates design and allows estimation of the performance of wireless systems under conditions different from those originally planned; (iii) analytical tractability: the model should have a reasonable analytical and numerical complexity, trying to avoid as much as possible the appearance of improper integrals (with an integration interval of infinite length) in the probability functions.

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2. Channel fading

In a real-world terrestrial propagation environment, radio signals are affected by various types of obstacles, including human bodies, trees, buildings, mountains. The primary physical effects of radio propagation are reflection, refraction, and scattering.

Regarding the distance compared to the wavelength of the radio signal, the received signal amplitude variations are divided into three categories: (i) very slow changes in signal strength over long distances are called path loss, (ii) slow variations due to terrain variations and signal blockage between transmitter and receiver due to the presence of large obstacles, such as buildings or hills, are called shadowing, (iii) fast variations of the radio signal due to multipath propagation and relative movement between transmitter and receiver are called (multipath or small-scale) fading [2].

Small-scale fading describes the overlapping effect of signals arriving from different paths with different amplitudes, delays, and phases at the receiver, with a spatial difference less than the radio wavelength or a short time. Thus, this multipath propagation involves the reception of radio signals from two or more paths at the receiving antennas. The combination of the received signal copies can be constructive or destructive, depending on the relative amplitudes, phases, and angles of arrival of the received signal components. The observed variation in the received overall signal amplitude is called multipath fading, which is sometimes used interchangeably with small-scall fading, or simply fading. The modeling of this kind of signal variations is the scope of this chapter.

Theoretically, the exact mathematical expression of channel fading can be determined by means of the Maxwell equations. However, due to the enormous complexity of an exact description of the physical propagation environment, statistical models are usually used instead when an analytical performance evaluation of a communication system is required. Currently, several statistical models have been proposed considering different propagation environments.

The effect of fading in wireless environments can be divided into two categories: slow and fast, which are primarily due to the speed of the relative movement of the transmitter and receiver or the surroundings reflecting the objects. In the frequency domain, fading also falls into two categories: flat fading and frequency selective. If the attenuation covers only a portion of the signal bandwidth, the fading is called frequency selective, and if the desired signal bandwidth is narrower than the spectrum affected by the fading, the fading is called flat fading.

In the design of telecommunication networks, the communication system designer must be able to predict the effects of multipath fading and noise on the mobile channel. A description of the most relevant classical and generalized fading models is presented in the following section.

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3. Small-scale fading models

In this section, the classic models of fading and some of the new models of fading will be reviewed.

3.1 Classical fading models

Classical fading models represent the baseband signal, h, as a complex Gaussian random variable, i.e.,

h=X+jY,E1

where X and Y are independent Gaussian random variables such that X:Nmxσx and Y:Nmyσy. Usually, the statistics of the envelope ∣h∣ are required or, more commonly, the power envelope (or equivalently, the SNR), γ=h2. The most commonly needed statistical functions are the cumulative distribution function (CDF), Fγx, the probability distribution function (PDF), fγx, and the moment generating function (MGF) Mγs=Eeγs, where E⋅ denotes the expectation operator. Each of these functions fully characterizes the power distribution, but depending on the metric to be computed, one of them may be the most convenient to use. Although, in theory, each of these functions can be obtained from each of the other two, in practice, one of the functions may have a very simple form, while the other(s) may have a difficult form or even not exist in closed-form.

The different classical fading models are obtained by setting different values for the parameters mxmyσxσy. As we now show, there are relevant recent new results for some of the models.

3.1.1 Rayleigh

This model is used when there is no direct line-of-sight (LOS) between the transmitter and receiver. It is by far the most commonly used model because of its analytical simplicity, since, for this model, the distribution of received power follows an exponential distribution. This model is obtained by setting the parameters mx=my=0 and σx=σy=σ. The PDF, CDF, and MGF of the SNR γ in this fading model are given by

fγx=12σ2e−x2σ2,E2
Fγx=1−e−x2σ2,E3
Mγs=11−2σ2s.E4

3.1.2 Rician

In this case, the nonzero mean of the Gaussian random variables X and/or Y model the presence of a direct visibility component between the transmitter and receiver (LOS), called the specular component. The Rician fading is the most commonly used model in this situation. However, it is much more difficult to handle analytically than the Rayleigh model, since special transcendental functions appear in its PDF (Bessel function) as well as in its CDF (Marcum Q function). The parameters are adjusted as σx=σy=σ, mx≠0 and/or my≠0. The PDF, CDF, and MGF of this model are

fγx=12σ2e−x+ν22σ2I0νxσ2,E5
Fγx=∑k=0∞ν22σ2ke−ν22σ2γk+1x2σ2,E6
Mγs=eν2s1−2σ2s1−2σ2s,E7

where ν=mx2+my2 and K=ν22σ2 are the channel parameters, I0. refers to the modified Bessel function of the first kind of order zero and γ.. is the lower incomplete Gamma function.

3.1.3 Hoyt (Nakagami-q)

Hoyt model is also an NLOS model and implies a greater variability of the signal (and therefore deeper fading) with respect to the Rayleigh channel. Its analytical treatment is more complex than that for the Rayleigh model, for the same reasons as for the Rice channel. The parameters are adjusted by σx≠σy, mx=my=0. It is interesting to note that the CDF was first given in closed-form in 2009 [3]. It is difficult to justify theoretically, in terms of the physics of propagation, why the in-phase and quadrature components should have a different variance (beyond a possible I/Q imbalance at the receiver). However, it has recently been shown that this model can be obtained from a Rayleigh fading model where the average power is not constant but varies in a certain way [4], i.e., the Hoyt model can be obtained from a Rayleigh model with a continuously varying average. The PDF, CDF, and MGF of the SNR of this fading model are given by

fγx=1+q22qγ¯e−1+q22x4q2γ¯I01−q4x4q2γ¯,E8
Fγx=2q1+q2−11−k2Qa1+q22x4q2γ¯b1+q22x4q2γ¯−Qb1+q22x4q2γ¯a1+q22x4q2γ¯,E9
Mγs=1−2γ¯s+q22γ¯s21+q22−12,E10

where I0. refers to the modified Bessel function of the first kind of order zero, Q.. is the first order Marcum Q function [4] and parameters are defined as q=minσxσymaxσxσy, γ¯=σx2+σy2, k=1−q21+q2, a=1+1−k2, b=1−1−k2, and x2=1+q22x4q2γ¯.

3.1.4 Beckmann

This is the most general case among the Gaussian-based classical fading models and includes all the previous models as special cases, where the parameters mx, my, σx, and σy take arbitrary values. However, it is hardly used because it does not have PDF and CDF in closed-form (i.e., in terms of known mathematical functions), although the MGF is given in the closed-form. Recently, a contribution has been presented in which this problem can be ignored, having been able to find different metrics of the wireless system in an exact way for this channel model [5]. Moreover, in this paper, for the first time, all moments of the distribution of the received power have been derived.

3.2 General new fading models

The received signal power may fluctuate significantly due to small-scale fading [6]. The summation of waves of constant amplitude with independent random phases provides a practical mathematical description of wave propagation [7, 8, 9]. The combined baseband voltage observed at the receiver has the form

V¯=∑i=1LViejΦi.E11

where Vi is the amplitude of the ith wave and Φi is the corresponding phase, which is assumed to be a random variable following a uniform distribution in the interval 02π [10]. The distances that radio waves travel are typically greater than the wavelength of the carriers, and the phases at each point in a small region are independent and unpredictable [11, 12]. Thus, wave propagation results in a series of waveform voltages at the terminals of an antenna [13]. In a small area, these voltages are mostly homogeneous plane waves, and their sum can be written as in (11) [14], which consists of the following terms.

  • Specular component: dominant wave represented, with baseband signal representation denoted as ViexpΦi.

  • Nonspecular component: nondominant wave whose power is insignificant compared to the average power of the entire scattered component.

  • The scattered (dispersed or diffuse) component is the aggregation of all the nonspecular components.

In a local area, the channel model can be written as an aggregate of a limited number of dominant and nondominant waves:

V˜=∑i=1NViejΦi⏟dominant component+∑j=N+1LViejΦi⏟non‐dominant component⏟dispersed componentE12

where N is the number of dominant waves. The grouping in the above equation is arbitrary, and any number of dominant components can, in principle, be considered. The nondominant components number L is unknown and the aggregation of these nondominants components V˜dif represents the diffuse received signal component due to the combined reception of numerous weak, independently phased scattered waves, which, by means of the Central Limit Theorem, can be statistically modeled by a complex Gaussian random variables, i.e.,

V˜=∑i=1NViejΦi+V˜dif,E13
=∑i=1NViejΦi+X+jY,E14

where X and Y are independent Gaussian random variables with zero mean of zero and variance σ2. Eq. (14) is the basis for some of the existing generalized fading channel models [1].

This model has been analyzed in all its generality in Ref. [15], and it has been shown that, for N>2, none of the main statistical functions (PDF, CDF, and MGF) can be expressed in closed-form.

3.2.1 Nakagami-m

This model has been widely used in the literature (so that it can be considered a “classical” model in a sense) and was initially presented as an empirical fit to real-world measurements. However, it can also be derived analytically. The received power of a Nakagami-m fading follows a Gamma distribution. Thus, the power can be obtained as the sum of m exponential (i.e., squared Rayleigh) random variables, and then the parameter m is allowed to take any arbitrary value once the distribution is obtained in order to improve the fit of the empirical measurements. Therefore, this model is based on underlying complex Gaussian random variables (at least for integer m). It is important to note that the diversity order of this model is equal to m. A single-input-single-output (SISO) channel always has a diversity order of 1. For this reason, the Nakagami-m model does not approximate the Rice model (which, like all classical models, has a diversity order of 1), especially in the tail of the distribution (which is of interest in communications theory). The Nakagami-m model has often been used to approximate the Rice model (since it is less analytically complicated). The PDF, CDF, and MGF of SNR of this model are

fγx=mmΓmΩmxm−1e−mΩx,E15
Fγx=γmmΩxΓm,E16
Mγs=1−Ωms−m,s<mΩ,E17

where m and Ω are channel parameters, Γ. is Gamma function and γ.. is the lower incomplete Gamma function.

3.2.2 Rician shadowed

The Rician shadowed (RS) fading model [16] is a generalization of the Rice model that allows the specular component to be subject to fluctuations modeled by the Nakagami-m distribution. These fluctuations have a clear physical sense, as various electromagnetic disturbances can affect the specular component, including ionospheric scintillation (for satellite communications), sudden changes in the electromagnetic field due to natural (e.g., solar activity) or artificial (e.g., motors, electric generators) causes, and so on. This model introduces an additional degree of freedom with respect to Rice, allowing for a better fit to field measurements, as

V˜=ζ+X+jY,E18

where ζ is a Nakagami-m random variable with mΩ parameters and X and Y are independent Gaussian random variables with a mean zero mean and variance of b0. The PDF, CDF, and MGF of SNR as γ experiencing this fading are

fγx=mm+Ω2b0me−x2b02b01F1m1Ω2b0x2b0m+Ω,E19
Fγx=1−2b0m2b0m+Ωm−1e−mx2b0m+Ω∑k=0mΩ2b0mkLm−k−1k−Ω2b0x2b0m+Ω,E20
Mγs=2b0mm1+2b0sm−12b0m+Ω1+2b0s−Ωm,E21

where 1F1... refers to the confluent hypergeometric function, also known as Kummer’s function or the confluent hypergeometric series. Lnν. is a Laguerre polynomial. The CDF was reported in Ref. [17] and it is valid for integer m.

3.2.3 Weibull

Another model that has gained some popularity, although smaller than the Nakagami-m model, is the Weibull model [18], which is obtained by a nonlinear transformation of the signal envelope (i.e., the square root of the power) of the Rayleigh model. The PDF and CDF, given in Refs. [19, 20], can be written as

fγx=m2Ωxm2−1e−1Ωxm2,E22
Fγx=1−e−1Ωxm2,E23

where m and Ω are the channel parameters. The MGF for 0<m<2 is given as

Mγs=m2Ω∑j=0q−1−1jj!Ωjsm2+mj2Γm2+mj2p+1Fq1Δpm2+mj2Δ(1j+1)−1qppspqqΩq,E24

and for m>2 is written as

Mγs=1Ω∑h=0p−1−shΩ1+chh!Γ1+chq+1Fp1Δq1+chΔ(p1+h)−spqqΩqppE25

with m=2pq and Δka=aka+1k…a+k−1k where p≥1 and q≥1 are coprime integers [20], and pFqa1…apb1…bbqz is known as a generalized hypergeometric function.

3.2.4 κ−μ and η−μ

Thr κ−μ and η−μ models [21] are multicluster models, which assume that various signal reflections arrive at the receiver grouped into sets or clusters, each of which follows a Rice distribution, for the κ−μ model, or Hoyt, for the η−μ model. These models allow the number of clusters to be an arbitrary real value, not necessarily a whole number, which makes it possible to model certain nonidealities, such as the correlation between clusters, that the central limit theorem is not verified in each cluster, or that the reflection of the waves does not occur at specific points, but on surfaces that may be rough, causing part of the energy to be scattered. As mentioned above, the Nakagami-m fading model can be interpreted in a similar way, where each cluster follows a Rayleigh fading and the clusters are added at the receiver. The PDF, CDF, and MGF are

fγx=μ1+κμ+12κμ−12eμκxμ−12e−μ1+κxIμ−12μκ1+κx,E26
Fγx=1−Qμ2κμ21+κμx,E27
Mγs=1+κμ1+κμ−sμe−κμ+κ1+κμ21+κμ−s.E28

where κ and μ are the channel parameters, Iμ. refers to the modified Bessel function of the first kind, and Qμ.. is the Generalized Marcum Q function [21].

3.2.5 κ−μ shadowed (KMS)

The κ−μ shadowed model [22] is an extension of the κ−μ model that allows the specular components of each cluster to fluctuate according to a Nakagami-m distribution, adding an additional degree of freedom compared to the κ−μ model. This additional degree of freedom makes the analysis less complex than for κ−μ. Moreover, it has been shown in Ref. [23] that under certain conditions, which usually occur in practice, the KMS model has an analytical complexity similar to Nakagami-m for m integer, so that it is possible to find closed-form expressions for numerous metrics of wireless systems (error rates, outage probability, channel capacity, etc.). The PDF, CDF, and MGF are

fγx=μμmm1+κμΓμγ¯μκ+mmxγ¯μ−1e−μ1+κxγ¯1F1mμμ2κ1+κμκ+mxγ¯,E29
Fγx=μμ−1mm1+κμΓμμκ+mmxγ¯μΦ2μ−mμμ+1−μ1+κxγ¯−μm1+κμκ+mxγ¯,E30
Mγs=−μμmm1+κμγ¯μμκ+mms−μ1+κγ¯m−μs−μ1+κγ¯mμκ+mm,E31

where κ, μ, m, and γ¯ are the channel parameters, 1F1... refers to the confluent hypergeometric function, also known as Kummer’s function or the confluent hypergeometric series, an Φ2. is the bivariate confluent hypergeometric function defined in Ref. [24].

3.2.6 α−μ, α−η−μ, and α−κ−μ

The α−μ fading model [25] explores the nonlinearity of the propagation medium, which is a rewritten form of the Stacy or generalized Gamma distribution. This distribution includes several others, such as Gamma and its discrete versions Erlang and central chi-squared, Nakagami-m and its discrete version chi, exponential, Weibull, one-sided Gaussian, and Rayleigh. Higher-order statistics for this model in closed-form formulas are given in Ref. [25].

The α−η−μ and α−κ−μ distributions include the α−μ, η−μ, κ−μ, Nakagami-m, Weibull, Rice, Hoyt, Rayleigh, and one-sided Gaussian as special cases. Field measurements are used to validate the distributions, and the moments, CDF, and estimators are derived in Ref. [26] for both the distributions.

The PDF and CDF for α−μ are

fγx=αμμ2Γμxαμ−22e−μxα2E32
Fγx=Γμμxα2ΓμE33

and, for the α−κ−μ , PDF and CDF are presented as

fγx=12ακ1−μ21+κ1+μ2μxα1+μ4−1e−μκ+1+κxα2Iμ−12κ1+κμxα4E34
Fγx=1−Qμ2μκxα42μ1+κE35

where α, κ, and μ are the channel parameters, Iμ. refers to the modified Bessel function of the first kind, and Qμ is the Generalized Marcum Q function.

3.2.7 Fluctuating Beckmann

The Fluctuating Beckmann (FB) [27] fading model is an extension and natural generalization of both the κ−μ shadowed and the classical Beckmann fading models. This model considers the clustering of multipath waves on which the LOS components fluctuate randomly, together with the effect of the in-phase/quadrature power imbalance in the LOS and NLOS components. Thus, FB unifies the one-sided Gaussian, Rayleigh, Nakagami-m, Rician, κ−μ, η−μ, Beckmann, Rician shadowed, and the κ−μ shadowed distributions as special cases.

The main probability functions of the FB fading model, namely PDF, CDF, and MGF, are derived in Ref. [27] as

fγx=α2m−m2xμ−1γ¯μΓμα1mΦ24−m+μ2−m+μ2mmμ−xγ¯ηα2−xηα2−xc1γ¯−xc2γ¯,E36
Fγx=α2m−m2xμγ¯μΓμ+1α1mΦ24−m+μ2−m+μ2mmμ+1−xγ¯ηα2−xηα2−xc1γ¯−xc2γ¯,E37

and

Mγs=11−2ημ1+η1+κγ¯sμ21−2μ1+η1+κγ¯sμ2×1−1mμκρ21+ρ21+ηγ¯s1+η1+κμ−2ηγ¯s+μκ11+ρ21+ηγ¯s1+η1+κμ−2γ¯s−m,E38

where α1=4ημ21+η21+κ2+2κρ2+ηm1+ρ2μ1+η1+κ2, α2=4ημ21+η21+κ2, β=−11+κ2μ+κm, and c1,2 are the roots of α1s2+βs+1. μ, m, κ, η, and γ¯ are channel model parameters and Φ24 is the confluent hypergeometric function.

3.2.8 TWDP

The Two-Wave with Diffuse Power (TWDP) [1] fading model extends the Rice model by considering that there are two specular components with independent phases that are added to a diffuse component modeled with a complex Gaussian distribution due to numerous weak and indistinguishable signal components. Despite the simplicity of its physical description, this model does not have its PDF and PDF in closed-form, which, in principle, somewhat limits its analytical applicability. Nevertheless, closed-form expressions for various metrics of wireless systems under this model have recently been found in Ref. [28]. The MGF, on the other hand, has recently been found in closed-form in Ref. [29]. Note that this model corresponds to Eq. (14) for the case N=2. The PDF is given by

fγx=e−x2σ2−K2σ2π∫0πeKΔcosθI02Kxσ21−Δcosθdθ≈e−x2σ2−K2σ2∑i=1MaiDxKΔcosπi−12M−1,E39

where

DxKαi=12eαiKI0x2K1−αi,+12e−αiKI0x2K1+αiE40

and αi=cosπi−12M−1 are defined in Ref. [29], and the MGF is

Mγs=1+K1+K−sγ¯eKsγ¯1+K−sγ¯I0Ksγ¯Δ1+K−sγ¯.E41

where K, Δ, and γ¯ are the channel model parameters and I0. refers to the modified Bessel function of the first kind of order zero.

3.2.9 FTR

The Fluctuating Two-Ray (FTR) [30] model extends the TWDP model by allowing both specular components to fluctuate according to a Nakagami-m distribution. The added degree of freedom allows, in contrast to the model from which it is derived, i.e., TWDP, to find closed-form expressions of the PDF and CDF. In addition, approximate (asymptotically exact) expressions of these functions have been found with a complexity similar to that of the Nakagami-m model, which is relatively simple to handle. It has been shown that the FTR model agrees well with experimental data from wireless systems in the millimeter-wave band (close to 30 GHz). Moreover, it is noteworthy that the FTR model includes the main classical models (Rayleigh, Rice, Hoyt) and even the Nakagami-m model as particular cases. The PDF, CDF, and MGF of the FTR are

fγx=12m−1K+1γ¯mm+K2−K2Δ2m∑q=0m−1/2−1qCqm−1m+Km+K2−K2Δ2m−1−2qΦ241+2q−mm−q−12m−q−121−m1−m1+Kxm+Kγ¯−m1+Kxm+K1+Δγ¯−m1+Kxm+K1−Δγ¯−1+Kxγ¯,E42
Fγx=12m−1K+1γ¯mm+K2−K2Δ2m∑q=0m−1/2−1qCqm−1m+Km+K2−K2Δ2m−1−2qxΦ241+2q−mm−q−12m−q−121−m2−m1+Kxm+Kγ¯−m1+Kxm+K1+Δγ¯−m1+Kxm+K1−Δγ¯−1+Kxγ¯,E43

and

Mγs=mm1+K1+K−γ¯sm−1RmKΔsmPm−1m1+K−m+Kγ¯sRmKΔs,E44

where Φ24 is the confluent hypergeometric function and RmKΔs is a polynomial in s defined as

RmKΔs=m+K2−Δ2K2γ¯2s2−2m1+Km+Kγ¯s+m21+K2E45

and Pμ. is the Legendre function of the first kind of degree μ, which can be calculated as Pμz=1F1−μμ+111−z2 and we have Cqn=2n−2q!q!n−q!n−2q!. After the appearance of the FTR, Zhang et al. [31] presented infinite series representations of the PDF and CDF for arbitrary fading parameters using a mixture of Gamma distributions as

fγx=mmΓm∑j=0∞Kjj!xje−x2σ2Γj+12σ2j+1∑k=0jjk∑l=0kklm+Kj+m+2l−k−KkKΔ22l×Rj+mk−2lKΔm+K2,E46
Fγx=mmΓm∑j=0∞Kjj!γj+1x2σ2Γj+1∑k=0jjk∑l=0kklm+Kj+m+2l−k−KkKΔ22l×Rj+mk−2lKΔm+K2,E47

where

Rνμx=ν−μ2ν−μ+12xμμ!×2F1ν+μ2ν+μ+121+μx,μ∈N+2F1ν−μ2ν−μ+121−μxΓ1−μ,otherwiseE48

In addition, Olyaee et al. [32] introduced two alternative frameworks for the statistical characterization and performance evaluation of the FTR fading model. The FTR fading distribution can be described, for arbitrary m, as an underlying Rician shadowed (RS) distribution with continuously varying parameters Kr (ratio of specular to diffuse power components), while for the special case of m being an integer, it is shown that the FTR fading model can be described in terms of a finite number of underlying squared Nakagami-m distributions. It was shown that any performance metric computed by averaging over the PDF of the FTR fading model can be expressed in terms of a finite-range integral over the corresponding performance metric for the simpler RS (for arbitrary m) or Nakagami-m (for integer m) fading models, many results of which are available in closed-form [32]. The PDF and CDF can be written as

fγxγ¯mKΔ=1π∫0πmm+K1+Δcosθmexp−1+Kγ¯x×1+Kγ¯1F1m11+KK1+Δcosθγ¯m+γ¯K1+Δcosθxdθ,E49
FγxmKΔ=1π∫0π1+Kγ¯xmm+K1+Δcosθm×Φ21−mm2−1+Kγ¯x−1+Kmxγ¯m+K1+Δcosθdθ,E50

where 1F1... refers to the confluent hypergeometric function, also known as Kummer’s function or the confluent hypergeometric series, an Φ2. is the bivariate confluent hypergeometric function defined in Ref. [24].

3.2.10 MTW

The Multicluster Two-Wave (MTW) [33, 34] fading model is a natural generalization of Durgin’s Two-Wave with Diffuse Power (TWDP) fading model by allowing the incident waves to arrive in different clusters. The MTW generalizes both the TWDP and the κ−μ models under a common umbrella. According to this model, multiple cluster of waves are independently received, with all of them potentially containing a specular component, except one, typically the first one received, which contains two specular components. This model has physical meaning in rich scattering environments, giving rise to many propagation paths, including a line-of-sight in the transmitter–receiver link, resulting in the aforementioned two dominant components in the first received cluster. The PDF, CDF, and MGF of the MTW model are given by

fγx=μπ1+Kγ¯μ+12e−μKxKμ−12e−μ1+Kγ¯x∫0πe−μKΔcosθ1+Δcosθ1−μ2×Iμ−12μ1+Kγ¯K1+Δcosθxdθ.E51
Fγx=1−1π∫0πQμ2μK1+Δcosθ2xμζdθ,E52

where ζ≜1+Kγ¯ and Qνab is the v-th order Generalized Marcum Q function as defined as ([18], eq. (4.60))

Qνab=a1−ν∫b∞xνexp−x2+a22Iν−1axdx,E53

and

Mγs=μ1+Kμ1+K−γ¯sμexpμKγ¯sμ1+K−γ¯sI0μKΔγ¯sμ1+K−γ¯s.E54

3.2.11 MFTR

The Multicluster Fluctuating Two-Ray (MFTR) [35] fading channel is modeled as the merging of two fluctuating random-phase specular components and a cluster of scattered waves. The MFTR model emerges as a natural generalization of both the FTR and the κ−μ shadowed fading models through a more general yet equally mathematically tractable model. It can also be seen as a generalization of the MTW model, where, by the addition of a new parameter, the specular components are allowed to fluctuate as a Nakagami-m-fading. The PDF, CDF, and MGF of the MFTR model are given by

fγx=1+Kμμμ2m−1Γμγ¯μmm+μK2−μ2K2Δ2m∑q=0m−12−1qCqm−1m+μKm+μK2−μ2K2Δ2m−1−2qxμ−1Φ241+2q−mm−q−1/2m−q−1/2μ−mμ−1+Kmμm+μKγ¯x−1+Kmμm+μK1+Δγ¯x−1+Kmμm+μK1−Δγ¯x−1+Kμγ¯x,E55
Fγx=1+Kμμμ2m−1Γμγ¯μmm+μK2−μ2K2Δ2m∑q=0m−12−1qCqm−1m+μKm+μK2−μ2K2Δ2m−1−2qxμΦ241+2q−mm−q−1/2m−q−1/2μ−mμ+1−1+Kmμm+μKγ¯x−1+Kmμm+μK1+Δγ¯x−1+Kmμm+μK1−Δγ¯x−1+Kμγ¯x,E56

and

Mγs=mmμμ1+Kμμ1+K−γ¯sm−μRμmKΔsmPm−1mμ1+K−μK+mγ¯sRμmKΔs,E57

where RμmKΔs is a polynominal in Ref. [35]. For m, a positive integer, the PDF and CDF of the SNR of the MFTR distribution are presented as an infinite discrete mixture of Gamma distributions as

fγx=∑j=0∞AjMFTRfGxj+1γ¯1+K,E58
Fγx=∑j=0∞AjMFTRFGxj+1γ¯1+K,E59

where AjMFTR is defined in Ref. ([35], eq. (18)), fGxj+1γ¯1+K and FGxj+1γ¯1+K are the PDF and CDF of the Gamma distribution in Ref. ([35], eq. (19)), and γ.. represents the incomplete Gamma function defined in Ref. ([17], eq. (6.5.2)).

3.2.12 IFTR

The Independent Fluctuating Two-Ray (IFTR) [36] fading model consists of two specular components that fluctuate independently, plus a diffuse component modeled as a complex Gaussian random variable. The IFTR model complements the FTR model, in which the specular components are fully correlated and fluctuate jointly. The main probability functions of the received SNR in IFTR fading, including the PDF, CDF, and MGF, can be expressed in closed-form. Also, a series form can be obtained for its probability functions in Ref. [37].

The importance of the IFTR model is that it is empirically validated using multiple channels measured in quite different scenarios, including LOS millimeter-wave, land mobile satellite (LMS), and underwater acoustic communications (UAC), and shows a better fit than the original FTR model and other models previously used in these environments [36]. For channel parameters m1, m2, K, Δ, and γ¯, the PDF, CDF, and MGF of SNR are presented in the following. The MGF of the SNR γ is written as

Mγs=1+K1+K−γ¯sm1m1m1−K21+1−Δ2γ¯s1+K−γ¯sm1m2m2m2−K21−1−Δ2γ¯s1+K−γ¯sm2×2F1m1m21K2Δ22m11+K−γ¯sγ¯s−K1+1−Δ212m21+K−γ¯sγ¯s−K1−1−Δ2,E60

where 2F1⋅ is the Gauss hypergeometric function. For m1∈Z+, the MGF of the SNR γ of the IFTR fading channel can be expressed as a finite sum of elementary terms as

Mγs=1+K1+K−γ¯sm1m1m2m2m1−K21+1−Δ2γ¯s1+K−γ¯sm2−m1∑n=0m1−11n!×m1−1nΓm2+nΓm2KΔ2γ¯s1+K−γ¯s2nm1m2−m1K21−1−Δ2+m2K2×1+1−Δ2γ¯s1+K−γ¯s−m2−n.E61

For m1∈Z+, the PDF and CDF of the SNR γ in an FTR fading channel are given by

fγx=1+Kγ¯m1m1m2m2m1+K2Δ2m2−m1∑n=0m1−11n!m1−1nΓm2+nΓm2KΔ22nm1K2Δ1+m2K2Δ2+m1m2−m2−nΦ23n+1−m1m1−m2m2+n1−1+Kγ¯x−m11+Km1+K2Δ2γ¯x−m1m21+Km1K2Δ1+m2K2Δ2+m1m2γ¯x,E62

and

Fγx=1+Kγ¯m1m1m2m2m1+K2Δ2m2−m1∑n=0m1−11n!m1−1nΓm2+nΓm2KΔ22nm1K2Δ1+m2K2Δ2+m1m2−m2−nxΦ23n+1−m1m1−m2m2+n2−1+Kγ¯x−m11+Km1+K2Δ2γ¯x−m1m21+Km1K2Δ1+m2K2Δ2+m1m2γ¯x.E63

where Δ1=1−1−Δ2, Δ2=1+1−Δ2, and Φ23⋅ is the confluent hypergeometric function. Series forms of the PDF and CDF of the IFTR model for arbitrary values of m1 and m2 are derived in Ref. [37] as

fγx=∑j=0∞AjIFTRfGxj+1γ¯1+K,E64
Fγx=∑j=0∞AjIFTRFGxj+1γ¯1+K,E65

where Aj is defined in Ref. ([37], eq. 11), fGxj+1γ¯1+K and FGxj+1γ¯1+K are the PDF and CDF of the Gamma distribution in Ref. ([37], eqs. (6) and (7)).

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4. Conclusions

This chapter reviews the classical and generalized statistical small-scale fading models, including some of the newest proposals in this field. Most of these fading models are interconnected, and it is common that the generalized models include the classical ones as particular cases, as shown in Figure 1. In this figure, the arrows connect a channel model to another, one which is a special case by adjusting the model parameters. For instance, the IFTR model can collapse into the TWDP and Rician shadowed models. Consequently, all analytical performance metrics results associated with the IFTR model (such as channel capacity, the bit error rate for different modulations) include the Rician, as well as the Rayleigh model performance metrics results as special cases.

Figure 1.

New channel models are general cases of some previous model. These connections between two channel models show that the model in the bottom row of arrows can be achieved as a special case.

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Written By

Maryam Olyaee, Hadi Hashemi and Juan Manuel Romero Jerez

Reviewed: 07 March 2024 Published: 29 May 2024

© 2024 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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